Dynamic Ordering for Asynchronous Backtracking on DisCSPs

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1 Dynamic Ordering for Asynchronous Backtracking on DisCSPs Roie Zivan and Amnon Meisels, Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, , Israel No Institute Given Abstract. An algorithm that performs asynchronous backtracking on distributed CSP s, with dynamic ordering of agents is proposed, ABT DO. Agents propose reorderings of lower priority agents and send these proposals whenever they send ok? messages. Changes of ordering triggers different computation of Nogoods. The dynamic ordered asynchronous backtracking algorithm uses polynomial space, similarly to standard ABT. The ABT DO algorithm with three different ordering heuristics is compared to standard ABT on randomly generated DisCSP s. A Nogood triggered heuristic, inspired by dynamic backtracking, is found to outperform static order ABT by a large factor in both run-time and network load. 1 Introduction Distributed constraint satisfaction problems (DisCSPs) are composed of agents, each holding its local constraints network, that are connected by constraints among variables of different agents. Agents assign values to variables, attempting to generate a locally consistent assignment that is also consistent with all constraints between agents (cf. [Yokoo, 2000a,Solotorevsky et al., 1996]). To achieve this goal, agents check the value assignments to their variables for local consistency and exchange messages with other agents, to check consistency of their proposed assignments against constraints with variables owned by different agents [Bessiere et al., 2005]. Distributed CSPs are an elegant model for many every day combinatorial problems that are distributed by nature. Take for example a large hospital that is composed of many wards. Each ward constructs a weekly timetable assigning its nurses to shifts. The construction of a weekly timetable involves solving a constraint satisfaction problem for each ward. Some of the nurses in every ward are qualified to work in the Emergency Room. Hospital regulations require a certain number of qualified nurses (e.g. for Emergency Room) in each shift. This imposes constraints among the timetables of different wards and generates a complex Distributed CSP [Solotorevsky et al., 1996]. A search procedure for a consistent assignment of all agents in a distributed CSP (DisCSP ), is a distributed algorithm. All agents cooperate in search for a globally consistent solution. The solution involves assignments of all agents to all their variables and exchange of information among all agents, to check the consistency of assignments with constraints among agents.

2 Asynchronous Backtracking (ABT ) is one of the most efficient and robust algorithms for solving DisCSP s. ABT was first presented by Y okoo [Yokoo et al., 1998,Yokoo, 2000a] and was developed further and studied in number of later studies [Hamadi, 2001,Bessiere et al., 2001,Silaghi and Faltings Agents in the ABT algorithms perform assignments asynchronously against their current view of the systems state. The method performed by each agent is in general simple. Later versions of ABT use polynomial space memory and perform dynamic backtracking [Bessiere et al., 2001,Bessiere et al., 2005]. ABT can be performed without adding links to the original constraint network [Bessiere et al., 2005]. ABT is also more robust to message delays than sequential (synchronous) assigning algorithms [Zivan and Meisels, 2004]. The versions of asynchronous backtracking presented in all the above studies use a static priority order among all agents. In centralized CSP s, dynamic variable ordering is known to be an effective heuristic for gaining efficiency [Dechter, 2003]. Recent studies have shown that the same is true for algorithms which perform sequential (synchronous) assignments in Distributed CSP s [Nguyen et al., 2004,Brito and Meseguer, 2004]. These studies suggest heuristics of agent (variable) ordering and empirically show large gains in efficiency over the same algorithms performing in static order. These results are the basic motivation for exploring the possibilities in dynamic reordering of asynchronous backtracking. In [Yokoo, 1995] an asynchronous algorithm with dynamic order is presented. According to Yokoo [Yokoo, 2000a], the Asynchronous Weak Commitment (AW C) algorithm performs better than ABT. However, in order to be complete, AW C uses exponential space which makes it unpractical for solving hard instances of DisCSP s. In [Hamadi, 2001] the authors present a distributed ordering algorithm which enables ordering the agents according to the properties of the constraints graph. However once the order is determined the asynchronous backtracking algorithm is performed using the static order chosen. Silaghi et. al. [Silaghi et al., 2001] argue that reordering in asynchronous backtracking is possible if the number of reordering operations is finite. In order to perform asynchronous finite reordering operations [Silaghi et al., 2001] suggest that the reordering operation will be performed by abstract agents. The results presented in [Silaghi et al., 2001] show minor improvements to static order ABT. The present paper proposes a simple algorithm for dynamic ordering in asynchronous backtracking, ABT DO that uses polynomial space, as standard ABT. In the proposed algorithm the agents of the DisCSP choose orders dynamically and asynchronously. As done in sequential assigning algorithms [Brito and Meseguer, 2004,Nguyen et al., 2004], an order is changed according to the progress of the search procedure. Agents in ABT DO perform according to the current, most updated order they hold. Each order is timestamped according to agents assignment. The method of time-stamp for defining the most updated order is the same that is used in [Nguyen et al., 2004] for choosing the most updated partial assignment. A simple array of counters represents for each suggested order its priority according to the global search tree. Each agent can change the order of all agents with lower priority. An agent can make this order change each time it replaces its assignment. Having established a correct algorithm for dynamic variable ordering in ABT, one needs to investigate ordering heuristics. Surprisingly, some of the heuristics which are very effective in sequential assignments dynamic ordering algorithms, do not improve

3 the run-time of ABT. It turns out that ordering heuristics based on Dynamic Backtracking [Ginsberg, 1993] are very successful (see the Experimental Evaluation section). Distributed CSP s are presented in the following section. The asynchronous backtracking with dynamic ordering (ABT DO) algorithm is presented next followed by a correctness and completeness proof for ABT DO. The Experimental Evaluation section, compares ABT to ABT DO with several ordering heuristics. The experiments were conducted on randomly generated DisCSP s. 2 Distributed Constraint Satisfaction A distributed constraints network (or a distributed constraints satisfaction problem - DisCSP) is composed of a set of k agents A 1, A 2,..., A k. Each agent A i contains a set of constrained variables X i1, X i2,..., X ini. Constraints or relations R are subsets of the Cartesian product of the domains of the constrained variables. For a set of constrained variables X ik, X jl,..., X mn, with domains of values for each variable D ik, D jl,..., D mn, the constraint is defined as R D ik D jl... D mn. A binary constraint R ij between any two variables X j and X i is a subset of the Cartesian product of their domains; R ij D j D i. In a distributed constraint satisfaction problem DisCSP, the agents are connected by constraints between variables that belong to different agents [Yokoo et al., 1998,Solotorevsky et al., 1996]. In addition each agent has a set of constrained variables, i.e. a local constraint network. An assignment (or a label) is a pair < var, val >, where var is a variable of some agent and val is a value from var s domain that is assigned to it. A compound label is a set of assignments of values to a set of variables. A solution P to a DisCSP is a compound label that includes all variables of all agents, that satisfies all the constraints. Following all former work on DisCSPs, agents check assignments of values against non-local constraints by communicating with other agents through sending and receiving messages. The following assumptions are routinely made in studies of DisCSP s and are assumed to hold in the present study [Yokoo, 2000a,Bessiere et al., 2001]. 1. All agents hold exactly one variable. 2. The amount of time that passes between the sending of a message to its reception is finite. 3. Messages sent by agent A i to agent A j are received by A j in the order they were sent. 4. An agent can send messages to any one of the other agents. 2.1 Asynchronous Backtracking The Asynchronous Backtrack algorithm (ABT ) was presented in several versions over the last decade and is described here in accordance with the more recent papers [Yokoo, 2000b,Bessiere et al., 2005]. In the ABT algorithm, agents hold an assignment for their variables at all times, which is consistent with their view of the state of the system (i.e. there Agent view). When the agent cannot find an assignment consistent with its Agent view, it changes its view by eliminating a conflicting assignment

4 from its Agent view data structure. It then sends back a Nogood which is based on its former inconsistent Agent view and makes another attempt to assign its variable [Yokoo, 2000b,Bessiere et al., 2005]. The Asynchronous Backtrack algorithm ABT [Yokoo, 2000b,Bessiere et al., 2005], has a total order of priorities among agents. Agents hold a data structure called Agent view which contains the most recent assignments received from agents with higher priority. The algorithm starts by each agent assigning its variable, and sending the assignment to neighboring agents with lower priority. When an agent receives a message containing an assignment (an ok? message [Yokoo, 2000b]), it updates its Agent view with the received assignment and if needed replaces its own assignment, to achieve consistency. Agents that reassign their variable, inform their lower priority neighbors by sending them ok? messages. Agents that cannot find a consistent assignment, send the inconsistent tuple in their Agent view in a backtrack message (a N ogood message [Yokoo, 2000b]) and extract the assignment of the lowest priority agent in the inconsistent tuple, from their Agent view. In the simplest form of the ABT algorithm, the complete Agent view is sent as a Nogood. The Nogood is sent to the agent with the lowest priority whose assignment is included in the Nogood. Agents that receive a N ogood, check its relevance against the content of their Agent view. If the Nogood is relevant the agent stores it, and tries to find a consistent assignment. In any case, if the agent receiving the Nogood keeps its assignment, it informs the Nogood sender by resending it an ok? message with its assignment. An agent A i which receives a Nogood containing an assignment of agent A j which is not included in its Agent view, adds the assignment of A j to it s Agent view and sends a message to A j asking it to add a link between them i.e. inform A i about all assignment changes it performs in the future. The performance of ABT can be improved immensely by requiring agents to read all messages they receive before performing computation [Yokoo, 2000b,Bessiere et al., 2005]. This technique was found to improve the performance of ABT on the harder instances of randomly generated DisCSPs by a large factor [Zivan and Meisels, 2003,Brito and Meseguer, 2004]. Another improvement to the performance of ABT can be achieved by using the method for resolving inconsistent subsets of the Agent view, based on methods of dynamic backtrack (as in CBJ). A version of ABT that uses this method was presented in [Bessiere et al., 2005]. In [Zivan and Meisels, 2003] the improvement of ABT using this method over ABT sending its full Agent view as a nogood was found to be minor. In all the experiments in this paper we used a version of ABT which includes both of the above improvements. Agents read all incoming messages that were received before performing computation and nogoods are resolved, using the dynamic backtracking method. 3 ABT with Dynamic Ordering Each agent in ABT DO holds a Current order which is an ordered list of pairs. Every pair includes the ID of one of the agents and a counter. Each agent can propose a new order each time it replaces its assignment. An agent A i can propose an order according to the following rules:

5 1. Agents with higher priority than A i and A i itself, do not change priorities in the new order. 2. Agents with lower priority than A i, in the current order, can change their priorities in the new order but not to a higher priority than A i. The counters attached to each agent ID in the order list form a time-stamp. Initially, all time-stamp counters are zero and all agents start the same Current Order. Each agent that proposes a new order changes the order of the pairs in its ordered list and updates the counters as follows: 1. The counters of agents with higher priority than A i, according to the Current order, are not changed. 2. The counter of A i is incremented by one. 3. The counters of agents with lower priority than A i in the Current order are set to zero. Consider an example in which agent A 2 holds the following Current order: (1, 4)(2, 3)(3, 1)(4, 0)(5, 1). There are 5 agents A 1...A 5 and they are ordered according to their IDs. After replacing its assignment it changes the order to: (1, 4)(2, 4)(4, 0)(5, 0)(3, 0). In the new order, agent A 1 which had higher priority than A 2 in the previous order keeps its place and the value of its counter does not change. A 2 also keeps its place and the value of its counter is incremented by one. The rest of the agents, which have lower priority than A 2 in the previous order, change places as long as they are still located lower than A 2. The new order is A 4, A 5, A 3 for these agents and their counters are set to zero. In ABT, agents send ok? messages to their neighbors whenever they perform an assignment. In ABT DO, ok? messages are accompanied by an order. The order message includes the agent s new Current order. An agent which receives an order message must determine if the received order is more updated than its own Current order. It decides by comparing the time-stamps lexicographically. Since orders are changed according to the above rules, every two orders must have a common prefix of the agents IDs since the agent that performs the change does not change its own position and the positions of higher priority agents. In the above example the common prefix includes agents A 1 and A 2. Since the agent proposing the new order increases its own counter, when two different orders are compared, at lease one of the time-stamp counters in the common prefix is different between the two orders. The more updated order is the one for which the first different counter in the common prefix is larger. In the example above, any agent which will receive the new order will know it is more updated than the previous order since the first pair is identical, but the counter of the second pair is larger. When an agent A i receives an order which is more up to date than its Current order, it replaces its Current order by the received order. The new order might change the location of the receiving agent with respect to other agents (in the new Current order). In other words, one of the agents that had higher priority than A i according to the old

6 when received (ok?, (x j, d j) do: 1. add (x j, d j) to agent view; 2. remove inconsistent nogoods; 3. check agent view; when received (order, received order) do: 1. if (received order is more updated than Current order) 2. Current order received order; 3. remove inconsistent nogoods; 4. check agent view; when received (nogood, x j, nogood) do 1. if (nogood contains an agent x k with lower priority than x i) 2. send (nogood, (x i, nogood)) to x k ; 3. send (ok?, (x i, current value) to x j; 4. else if (nogood consistent with Agent view and current assignment) 6. store nogood; 7. if (nogood contains an agent x k that is not its neighbor) 8. request x k to add x i as a neighbor; 9. add (x k, d k ) to agent view; 10. check agent view; 11. else 12. send (ok?, (x i, current value)) to x j; Fig. 1. The ABT DO algorithm (first part) order, now has a lower priority than A i or vise versa. Therefore, A i rechecks the consistency of its current assignment and the validity of its stored N ogoods according to the new order. If the current assignment is inconsistent according to the new order, the agent makes a new attempt to assign its variable. In ABT DO agents send ok? messages to all constraining agents (i.e. their neighbors in the constraints graph). Although agents might hold in their Agent views assignments of agents with lower priorities, according to their Current order, they eliminate values from their domain only if they violate constraints with higher priority agents. A Nogood message is always checked according to the Current order of the receiving agent. If the receiving agent is not the lowest priority agent in the Nogood according to its Current order, it sends the Nogood to the lowest priority agent and sends an ok? message to the sender of the Nogood. This is a similar operation to that performed in standard ABT for any unaccepted Nogood. Figures 2 and 1 present the code of asynchronous backtracking with dynamic ordering (ABT DO).

7 procedure check agent view 1. if(current assignment is not consistent with all higher priority assignments in agent view) 2. if(no value in D i is consistent with all higher priority assignments in agent view) 3. backtrack; 4. else 5. select d D i where agent view and d are consistent; 6. current value d; 7. Current order choose new order 8. send (ok?,(x i, d)) to neighbors; 9. send (order,current order) to lower priority agents; procedure backtrack 1. nogood resolve inconsistent subset; 2. if (nogood is empty) 3. broadcast to other agents that there is no solution; 4. stop; 5. select (x j, d j) where x j has the lowest priority in nogood; 6. send (nogood, x i, nogood) to x j; 7. remove (x j, d j) from agent view; 8. remove all Nogoods containing (x j, d j); 9. check agent view; Fig. 2. The ABT DO algorithm(second part) When an ok? message is received (first procedure in Figure 2), the agent updates the Agent view and removes inconsistent N ogoods. Then it calls check agent view to make sure its assignment is still consistent. A new order received in an order message is accepted only if its more up to date than the Current order (second procedure of Figure 2). If so, the received order is stored and check agent view is called to make sure the current assignment is consistent with the higher priority assignments in the Agent view. When a Nogood is received (third procedure in Figure 2) the agent first checks if it is the lowest priority agent in the received Nogood, according to the Current order. If not, it sends the Nogood to the lowest priority agent and an ok? message to the Nogood sender (lines 1-3). If the receiving agent is the lowest priority agent it acts exactly the same as in the standard ABT algorithm (lines 4-12). Procedure backtrack (Figure 1) is the same as in standard ABT. The Nogood is resolved and the result is sent to the lower priority agent in the Nogood, according to the Current order. Procedure check agent view (Figure 1) is very similar to standard ABT but the small difference is substantial (lines 5-9). If the current assignment is not consistent and must be replaced and a new consistent assignment is found, the agent chooses a new order as its Current order (line 7) and updates the corresponding time-stamp. After the ok? messages are sent, the new order and its time-stamp counters are sent to all lower priority agents.

8 4 Correctness of ABT DO In order to prove the correctness of the ABT DO algorithm we first establish two facts by proving the following lemmas: Lemma 1 The highest priority agent in the initial order remains the highest priority agent in all the following proposed orders. The proof for Lemma 1 is immediate from the two rules of reordering. Since no agent can propose a new order which changes the priority of higher priority agents and its own priority, no agent including the first can move the highest priority agent to a lower position. Lemma 2 When the highest priority agent proposes a new order, it is more up to date than all previous orders. In all previous orders the time-stamp counter of the first agent is smaller than the counter of the time-stamp counter of the first agent in the new proposed order. To prove correctness of a search algorithm for DisCSP s one needs to prove that it is sound, complete and that it terminates. ABT DO, like ABT, reports a solution when all agents are idle and no messages are sent. Its soundness follows from the soundness of ABT [Bessiere et al., 2005]. One point needs mentioning. Since no messages are traveling in the system in the idle state, all overriding messages have arrived at their destinations. This means that for every pair of constraining agents an agreement about their pairwise order has been achieved. One of each pair of constraining agents checks that constraint and no messages mean no violations, as in the proof for ABT [Bessiere et al., 2005]. To prove the completeness and termination of ABT DO we use induction on the number of agents (i.e. number of variables) in the DisCSP. For a single agent DisCSP the order is static therefore the completeness and termination of ABT implies the same for ABT DO. Assume ABT DO is complete and terminates on every DisCSP with k agents where k < n. Consider a DisCSP with n agents. According to Lemma 1 the agent with the highest priority in the initial order will not change its place. The highest priority agent assigns its variable for the first time and sends it along with its order proposal to other agents. The remaining DisCSP has n 1 agents and its initial order is that proposed by the first agent (all other orders are discarded according to Lemma 2). By the induction assumption the remaining DisCSP is complete and terminates. If a solution to the induced DisCSP is found, the algorithm terminates and reports a solution. Otherwise, a single assignment N ogood will be sent to the highest priority agent which will cause it to replace its assignment. The new assignment of the first agent and the new order proposed will induce a new DisCSP of size n 1. The search on this new DisCSP of size n 1 is also complete and terminates according to the induction assumption. The number of induced DisCSP s, created by the assignments of the highest priority agent is bound by the size of its domain. Therefore, the algorithm will terminate in a finite time. The algorithm is complete since a solution to the DisCSP must include one of the highest priority agent value assignments which means that one of the induced DisCSP s includes a solution iff the original DisCSP includes a solution.

9 If the network model enables agents to communicate only with their neighbors in the constraints network, some small changes are needed in order to keep the algorithm correct. First, agents must be allowed to change only the order of lower priority neighbors. This means that the method choose new order called in line 7 of procedure check agent view changes the order by switching between the possition of lower priority neigbors and leaving other lower priority agent at their current position. Second, whenever a new order message is received, an agent informs its neighbors of its updated current order. In order to prove that the above two changes do not affect the correctness of the algorithm we first establish the correctness of Lemmas 1 and 2 under these changes. Lemma 1 is not affected by the change since the rules for changing agents positions have become more strict, and still not allow to change higher priority agents. Lemma 2 holds since the time-stamp mechanism which promices its correctness have not changed. These Lemmas are the basis for the correctness of the induction which proves the algorithm is complete and terminates. However, we still need to prove the algorithm is sound. One of the assumptions that our soundness proof dependent on was that an idle state of the system would mean that every constrained couple of agents agree on the order between them. This claim might not hold since the most up to date order is not sent to all agents. The following Lemma proves this claim is still true after the changes in the algorithm: Lemma 3 When the system reaches an idle state, every pair of constrained agents hold the same order. Whenever one of the constrained agents will receive an updated order message, it will inform its neighbors therefore all agents which have constraints with it will be notified and hold the updated order. If two agents are not informed with the most updated order, this would mean both of them are not lower priority neighbors of the reordering agents therefore their current position in the order stays the same. Lemma 3 implies that the algorithm is still sound according to our previous proof. 5 Experimental Evaluation The common approach in evaluating the performance of distributed algorithms is to compare two independent measures of performance - time, in the form of steps of computation [Lynch, 1997,Yokoo, 2000a], and communication load, in the form of the total number of messages sent [Lynch, 1997]. Non concurrent steps of computation, are counted by a method similar to that of [Lamport, 1978,Meisels et al., 2002]. Every agent holds a counter of computation steps. Every message carries the value of the sending agent s counter. When an agent receives a message it updates its counter to the largest value between its own counter and the counter value carried by the message. By reporting the cost of the search as the largest counter held by some agent at the end of the search, a measure of non-concurrent search effort that is close to Lamports logical time is achieved [Lamport, 1978]. If instead of steps of computation, the number of non concurrent constraints check is

10 Fig. 3. Non concurrent constraints checks performed by ABT and ABT DO using different order heuristics on low density DisCSPs (p 1 = 0.4). counted (NCCCs), then the local computational effort of agents in each step is measured [Meisels et al., 2002]. Experiments were conducted on random networks of constraints of n variables, k values in each domain, a constraints density of p 1 and tightness p 2 (which are commonly used in experimental evaluations of CSP algorithms [Smith, 1996]). All three sets of experiments were conducted on networks with 20 agents (n = 20) each holding exactly one variable, 10 values for each variable (k = 10) and two values of constraints density p 1 = 0.4 and p 1 = 0.7 The tightness value p 2, is varied between 0.1 and 0.9, to cover all ranges of problem difficulty. For each pair of fixed density and tightness (p1, p2) 50 different random problems were solved by each algorithm and the results presented are an average of these 50 runs. ABT DO is compared to the run of standard ABT. For ordering variables in ABT DO three different heuristics were used. 1. Random: each time an agent changes its assignment it randomly orders all agents with lower priorities in its Current order. 2. Domain-Size: This heuristic was inspired by the heuristics used for sequential assigning algorithms in [Brito and Meseguer, 2004]. Domain sizes are calculated based on the fact that each agent that performs an assignment sends its current domain size to all other agents. Every agent that replaces an assignment, orders the lower priority agents according to their domain size from the smallest to the largest. 3. Nogood-Triggered: Agents change the order of the lower priority agents only when they receive a N ogood which eliminates their current assignment. In this case the agent moves the sender of the Nogood to be in front of all other lower priority agents. This heuristic was inspired by the heuristic used for dynamic backtracking in centralized CSP s [Ginsberg, 1993].

11 Fig. 4. Total number of messages sent by ABT and ABT DO on low density DisCSPs (p 1 = 0.4). Fig. 5. Non concurrent constraints checks performed by ABT and ABT DO using different order heuristics on high density DisCSPs (p 1 = 0.7). Figure 3 presents the computational effort in number of non concurrent constraints checks to find a solution, performed by ABT and ABT DO using the above three heuristics. The algorithms solve low density DisCSP s with p 1 = 0.4. ABT DO with random ordering does not improve the results of standard ABT. ABT DO which uses the domain sizes to order the lower priority agents performs slightly better than ABT.

12 Fig. 6. Total number of messages sent by ABT and ABT DO on high density asymmetric DisC- SPs (p 1 = 0.7). The largest improvement is gained by using the N ogood trigerred heuristic. For the hardest DisCSP instances, ABT DO using the N ogood trigerred heuristic improves the performance of standard ABT by a factor of 5. Figure 4 presents the total number of messages sent by the algorithms for the same problems. While ABT DO with random ordering heuristic shows similar run time results to standard ABT it sends almost twice as many messages. This can be expected since in ABT D O agents send ok messages to all their neighbors while in standard ABT ok? messages are sent only to lower priority agnets. ABT DO with domain size ordering sends more messages than standard ABT but less than the random ordering version. The most economic in the use of the network is ABT DO with the Nogood triggered heuristic which sends three time less messages than standard ABT on the hardest DisCSP instances. Figures 5 and 6 present similar results for high density DisCSPs - p 1 = 0.7. It seams the influence of the ordering heuristics on the performance of the algorithm is not dependent on the network density. 6 Discussion Dynamic ordering is a powerful heuristic used to improve the run-time of centralized CSP algorithms [Dechter, 2003] and of distributed CSP algorithms with sequential assignments [Brito and Meseguer, 2004,Nguyen et al., 2004]. The results in the previous section, show that dynamic ordering must be combined with the right heuristic in order to improve the run-time and justify the overhead in message load of asynchronous backtracking. A random order heuristic does not improve the run-time of standard ABT and sends many more messages. Surprisingly, ordering the agents accord-

13 ing to their domain size does not gain a large improvement as reported for sequential (synchronous) assignments algorithms by [Brito and Meseguer, 2004]. This can be explained by the fact that asynchronous backtracking prunes the DisCSP search tree by storing N ogoods which prevent it from trying to extend inconsistent tuples. N ogoods are discarded in standard ABT whenever they become irrelevant [Bessiere et al., 2005]. In ABT DO, this can happen when an agent holding a Nogood changes places with one of the other agents whose assignment appears in the Nogood. On the other hand, the N ogood triggering heuristic, inspired by Dynamic Backtracking [Ginsberg, 1993] was found to be very effective. In this heuristic, the above example of losing useful information cannot occur. N ogoods are resolved and created according to dynamic backtracking. They include all the conflicting assignments held by the Nogood sender. An agent that is moved to a higher priority position in the order, is moved lower than all the agents with conflicting assignments, therefore no Nogoods are discarded. The results show that this heuristic is very effective in both measures, run-time and network load. The improvement in network load is particularly striking in view of the additional ordering messages of ABT DO. 7 Conclusions Most studies of asynchronous backtracking used a static order of agents [Hamadi, 2001,Yokoo, 2000a,Bessiere et al., 200 An exponential space algorithm using dynamic ordering has shown improvement in run-time over ABT [Yokoo, 2000a]. The only study that suggested dynamic ordering in ABT with polynomial space used a complex method including additional abstract agents [Silaghi et al., 2001]. The results presented in [Silaghi et al., 2001] show a minor improvement compared to standard, static order, ABT. The present study proposes a simple way of performing dynamic ordering in ABT with polynomial space. The ordering is performed as in sequential assignment algorithms by each agent changing only the order of agents following it in the former order. A simple method of time-stamping [Nguyen et al., 2004] is used to determine the most updated proposed order. When a heuristic order inspired by dynamic backtracking [Ginsberg, 1993] is used to dynamically reorder agents, there is a significant improvement in run-time and network load over standard ABT. References [Bessiere et al., 2001] C. Bessiere, A. Maestre, and P. Messeguer. Distributed dynamic backtracking. In Proc. Workshop on Distributed Constraint of IJCAI01, [Bessiere et al., 2005] C. Bessiere, A. Maestre, I. Brito, and P. Meseguer. Asynchronous backtracking without adding links: a new member in the abt family. Artificial Intelligence, 161:1-2:7 24, January [Brito and Meseguer, 2004] I. Brito and P. Meseguer. Synchronous,asnchronous and hybrid algorithms for discsp. In Workshop on Distributed Constraints Reasoning(DCR-04) CP-2004, Toronto, September [Dechter, 2003] Rina Dechter. Constraints Processing. Morgan Kaufman, 2003.

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